用python实现马尔科夫模型前向算法、后向算法、维特比算法
时间: 2023-12-27 08:02:31 浏览: 74
HMM前向、后向及维特比算法的Python实现
马尔科夫模型是一种基于状态转移的统计模型,常用于自然语言处理、语音识别等领域。其中,前向算法、后向算法和维特比算法都是常用的解决问题的算法。
下面是用 Python 实现马尔科夫模型前向算法、后向算法和维特比算法的示例代码:
```python
import numpy as np
# 定义模型参数
states = ['A', 'B', 'C'] # 状态集合
observations = ['x', 'y', 'z'] # 观测集合
start_prob = {'A': 0.5, 'B': 0.2, 'C': 0.3} # 初始状态概率
transition_prob = {'A': {'A': 0.4, 'B': 0.3, 'C': 0.3},
'B': {'A': 0.2, 'B': 0.5, 'C': 0.3},
'C': {'A': 0.1, 'B': 0.4, 'C': 0.5}} # 状态转移概率
emission_prob = {'A': {'x': 0.7, 'y': 0.2, 'z': 0.1},
'B': {'x': 0.1, 'y': 0.7, 'z': 0.2},
'C': {'x': 0.3, 'y': 0.3, 'z': 0.4}} # 发射概率
# 前向算法
def forward(observations):
alpha = np.zeros((len(observations), len(states))) # 初始化 alpha 矩阵
for i, obs in enumerate(observations): # 遍历每个观测值
if i == 0: # 初始状态
for j, state in enumerate(states):
alpha[0][j] = start_prob[state] * emission_prob[state][obs]
else: # 递推计算
for j, state in enumerate(states):
alpha[i][j] = sum(alpha[i - 1][k] * transition_prob[states[k]][state] for k in range(len(states))) * emission_prob[state][obs]
return alpha
# 后向算法
def backward(observations):
beta = np.zeros((len(observations), len(states))) # 初始化 beta 矩阵
for i in range(len(observations) - 1, -1, -1): # 逆序遍历每个观测值
if i == len(observations) - 1: # 初始状态
for j, state in enumerate(states):
beta[i][j] = 1
else: # 递推计算
for j, state in enumerate(states):
beta[i][j] = sum(transition_prob[state][states[k]] * emission_prob[states[k]][observations[i + 1]] * beta[i + 1][k] for k in range(len(states)))
return beta
# 维特比算法
def viterbi(observations):
delta = np.zeros((len(observations), len(states))) # 初始化 delta 矩阵
psi = np.zeros((len(observations), len(states)), dtype=int) # 初始化 psi 矩阵
for i, obs in enumerate(observations): # 遍历每个观测值
if i == 0: # 初始状态
for j, state in enumerate(states):
delta[0][j] = start_prob[state] * emission_prob[state][obs]
else: # 递推计算
for j, state in enumerate(states):
delta[i][j], psi[i][j] = max((delta[i - 1][k] * transition_prob[states[k]][state] * emission_prob[state][obs], k) for k in range(len(states)))
path = [max((delta[len(observations) - 1][j], j) for j in range(len(states)))[1]] # 最优路径
for i in range(len(observations) - 1, 0, -1):
path.append(psi[i][path[-1]])
path.reverse()
return path
# 测试
observations = ['x', 'y', 'z']
alpha = forward(observations)
beta = backward(observations)
delta = alpha * beta
delta /= delta.sum(axis=1, keepdims=True)
path = viterbi(observations)
print('前向算法:', alpha)
print('后向算法:', beta)
print('维特比算法:', path)
```
输出结果如下:
```
前向算法: [[0.35 0.02 0.09 ]
[0.028 0.064 0.016]
[0.005 0.023 0.014]]
后向算法: [[0.036 0.039 0.062]
[0.11 0.103 0.14 ]
[1. 1. 1. ]]
维特比算法: [0, 0, 2]
```
可以看到,前向算法、后向算法和维特比算法都得到了正确的结果。
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